Select using intersection or union?
Unfortunately, selection tools don't support this directly. However, there are ways to accomplish the same result. For example:


Deselect everything (A)
Select the vertices you want to "intersect" with the previous selection. In your case, the vertices in the green rectangle.
Hide the other geometry (&#8679; ShiftH)
Do whatever it is you did to get the initial selection that you want to intersect (e.g. select by material)
Create and assign a vertex group to store the selection
Unhide everything (&#9095; AltH) and deselect (A)
Select the vertex group by clicking Select in the vertex groups  panel.




However in most cases like yours the desired selection can easily be obtained with one or two box deselections. To deselect with box select (B), use  MMB or hold &#8679; Shift when defining the box.